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Problem 805

AMC 12 late, AIME early

Geometry Difficulty 4.4 Prove it Harvard-MIT Mathematics Tournament · United States

Let $x$ and $y$ be two distinct roots of unity. Prove that $x+y$ is also a root of unity if and only if $\frac{y}{x}$ is a cube root of unity.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

This is easiest to see geometrically. The vectors corresponding to $x$ , $y$ , and $-x-y$ sum to $0$ , so they form a triangle. In order for them all to be roots of unity, they must all have length one, so the triangle must be equilateral. Therefore the angle between $x$ and $y$ is $\pm \frac{2 \pi}{3}$ , that is, $\frac{y}{x}$ is a cube root of unity.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.